Nonlinear interactions in turbulence with strong irrotational straining

The rate of growth of the nonlinear terms in the vorticity equation are analysed for a turbulent flow with r.m.s. velocity u(0) and integral length scale L subjected to a strong uniform irrotational plane strain S, where (u(0)/L)/S = epsilon much less than 1. The rapid distortion theory (RDT) solution is the zeroth-order term of the perturbation series solution in terms of e. We use the asymptotic form of the convolution integrals for the leading-order nonlinear terms when beta = exp(-St) much less than 1 to determine at what time t and beyond what wavenumber k (normalized on L) the perturbation series in epsilon fails, and hence derive the following conditions for the validity of RDT in these flows. (a) The magnitude of the nonlinear terms of order epsilon depends sensitively on the amplitude of eddies with large length scales in the direction x(2) of negative strain. (b) If the integral of the velocity component u(2) is zero the leading-order nonlinear terms increase and decrease in the same way as the linear terms, even those that decrease exponentially. In this case RDT calculations of vorticity spectra become invalid at a time t(NL) similar to L/u(0)k(-3) independent of epsilon and the power law of the initial energy spectrum, but the calculation of the r.m.s. velocity components by RDT remains accurate until t = T-NL similar to L/u(0), when the maximum amplification of r.m.s. vorticity is omega/S similar to epsilon exp(epsilon(-1)) much greater than 1. (c) If this special condition does not apply, the leadingorder nonlinear terms increase faster than the linear terms by a factor O(beta(-1)). RDT calculations of the vorticity spectrum then fail at a shorter time t(NL) similar to(1/S)1n((epsilon-1)k(-3)); in this case T-NL similar to(1/S)1n(epsilon(-1)) and the maximum amplification of r.m.s. vorticity is omega/S similar to 1. (d) Viscous effects dominate when t much greater than (1/S) In(k(-1)(Re/epsilon)(1/2)). In the first case RDT fails immediately in this range, while in the second case RDT usually fails before viscosity becomes important. The general analytical result (a) is confirmed by numerical evaluation of the integrals for a particular form of eddy, while (a), (b), (c) are explained physically by considering the deformation of differently oriented vortex rings. The results are compared with small-scale turbulence approaching bluff bodies where epsilon much less than 1 and beta much less than 1. These results also explain dynamically why the intermediate eigenvector of the strain S aligns with the vorticity vector, why the greatest increase in enstrophy production occurs in regions where S has a positive intermediate eigenvalue; and why large-scale strain S of a small-scale vorticity can amplify the small-scale strain rates to a level greater than S - one of the essential characteristics of high-Reynolds-number turbulence.